(1). We prove the assertion for \(i=2\) ; the case when \(i=1\)  is similar.
After postcomposing with an element of \(\operatorname{PGL}(3,\mathbb {R})\) , may assume without loss of generality that \(f\equiv \operatorname{id}\)  on an open set \(V\subset U\)  with \((\bar{W}_1,\bar{W}_2)\in V\) .
Pick \((W_1,W_2)\in U\cap \pi _2^{-1}(\bar{W}_2)\) , i.e. \(W_2=\bar{W}_2\) . Choose \((W_1,W_2^{\prime })\in U\setminus \lbrace (W_1,W_2)\rbrace \)  such that \(\pi _2^{-1}(W_2^{\prime })\cap V\ne \emptyset \) .
Now suppose \((W_1^j,W_2^j)\rightarrow (W_1,W_2)\) . We may choose \(\lbrace W_2^{\prime j}\rbrace \subset 2\)  such that \(W_1^j\subset W_2^{\prime j}\)  and \(W_2^{\prime j}\rightarrow W_2^{\prime }\)  as \(j\rightarrow \infty \) . Since \(V\)  is open, after dropping finitely many terms, we may assume that \(\pi _2^{-1}(W_2^j)\cap V\ne \emptyset \)  and \(\pi ^{-1}(W_2^{\prime j})\cap V\ne \emptyset \) . Since \(f\)  is fibration-preserving and \(f\equiv \operatorname{id}\)  on \(V\) , it follows that \(f(\pi _2^{-1}(W_2^j)\cap U)\subset \pi _2^{-1}(W_2^j)\)  and \(f(\pi ^{-1}(W_2^{\prime j})\cap U)\subset \pi ^{-1}(W_2^{\prime j})\) . The map \(f\)  is fibration preserving, so \(f(W_1^j,W_2^j), f(W_1^j,W_2^{\prime j})\in \pi _1^{-1}(f_1(W_1^j))\) . Therefore \(\pi _1^{-1}(f_1(W_1^j))\)  intersects both \(\pi _2^{-1}(W_2^j)\)  and \(\pi _2^{-1}(W_2^{\prime j})\)  nontrivially, forcing \(f_1(W_1^j)=W_2^j\cap W_2^{\prime j}=W_1^j\) . Thus \(f(W_1^j,W_2^j)=(W_1^j,W_2^j)\) . Since the sequence \(\lbrace (W_1^j,W_2^j)\rbrace \)  was arbitrary, this proves (1).
(2). Define an equivalence relation on \(U\)  where \(x,x^{\prime }\in U\)  are equivalent if there is a path \(\gamma :[0,1]\rightarrow U\)  from \(x\)  to \(x^{\prime }\)  which is piecewise contained in a fiber of one of the fibrations \(\pi _i:i\) . We claim that the equivalences classes are open subsets of \(U\) . To see this, pick \(x=g\cdot (W_1^-,W_2^-)\in U\) , and note that the vector fields \(X_{1,2}\) , \(X_{2,3}\)  on \(N\)  are bracket generating and tangent to the cosets of \(K_1\)  and \(K_2\) , respectively; and hence their pushforwards \(X_{1,2}^{\prime }\) , \(X_{2,3}^{\prime }\)  under the composition \(N\stackrel{\alpha }{\rightarrow }\hat{N}\stackrel{g}{\rightarrow }g\cdot \hat{N}\)  are bracket generating vector fields on \(g\cdot \hat{N}\)  which are tangent to the fibers of \(\pi _1\)  and \(\pi _2\)  respectively, by Lemma REF (4). By [1]}
the equivalence class of \(x\)  contains a neighborhood of \(x\) . It follows that the equivalence classes are connected components of \(U\) . Since the set where \(f\)  locally agrees with \(g\)  is a union of equivalences classes by (1), assertion (2) follows.
